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In N. J. Smelser and P. B. Baltes, (Eds.), 2001. International
Encyclopedia of the S o c i a l & B e h a v i o r a l Sciences. Oxford:
Elsevier, 1026-1 032.
the subsequent discussion here, is that, already in the
next century, the fourth century BC, the detailed and
elaborate theory of proportion of Eudoxus emerged in
quite a clear and definite axiomatic form. most of
which is preserved in Book V of Euclid's Elenienrs
(Euclid 1925). What is important about E u d o ~ u ~ ' ~
work, and even the commentaries of the work of this
time, for example, by Aristotle, in the Poslerior
Ar~a!l-.!ics (1994 74a-17). is that the theorems were
proved, not for single geometric objects, but for
magnitudes in general, when applicable. The recog.
nition of the correct abstraction in the general
concept of magnitude, its technical and thorough
ilnplementation by Eudoxus and the philosophical
commentary by Aristotle, represent a genuinely new
intellectual development. The language of E u d o x u ~ ' ~
famous Definition V in Book V of Euclid's Elernetits matches in its abstractness and difficulty the
standards of modern axiomatic theories in mathenlatics and the sciences.
Definition 5 Mugr?i/~rdesnrc soid to be ill (lie snnie
r a t i o , thefirs1 10 the sacorid ond the third to //7e,/i7irrlh,
when, ifn17y eqrri17iultiples \~.lintei~er
be t ~ l k e t i~ f t h e f i r . ~ ~
atid third, arztl ony equitr~ul/il~les
~,t>hateuero/'~hesecond
or7d fourth, the f o r r ~ e r eqiritriultiples alike r.rceed, ore
alike eylral t o , or nlikr ,/irll slrort q / : the lrrtter equir7i~rltiplesresprcti~le(vtnkrrr iri ror.resporidirrg order.
Axiomatic Theories
Axiomatic development of theories is common practice in pure mathematics and is also now widely used in
many sciences. The main ingredients of the methods
for axio~natizingtheories are the following: statement
of the primitive concepts of the theory, statement of
the prior mathematics basis assumed. statement of the
axioms. characterization of niodels of the theory and a
definition of two such models having the same structure. Theories formulated in this way can easily satisfy
the standard set-theoretical approach t o axiomatization. The further step of formalizing the language of
the theory, especially in the case of elementary
theories, can lead to specific positive and negative
results about the axioniatizability of theories in restricted languages.
The codification of the Greek axiomatic approach
in Euclid's Elerrients was a great success and remained
almost unchallenged until difficulties in the dekails
were found in the eighteenth and nineteenth centuries,
as discussed later.
Various scientific examples of axiomatic theories
existed already in ancient Greek times and, of course,
from the Greek standpoint. they were regarded as
essentially homogeneous with the axioniatic theory of
geometry, no sharp distinction being made between
geometry and mechanics, for instance. A good example is Archimedes' set of partial qualitative axioms
For measuring weights on balances. This is undoubtedly the first partial qualitative axioma~izationof
conjoint measurement, a form of measurenient that
has received both much axioniatic attention and
manifold applications in modern theories of measurement in the social sciences. (For detailed treatment of
conjoint measurement, see Krantz et al. 197 1. Alsosee
Measurer?ient T l i e o y : Cocjoirrt.)
I.
Historical Backgro~117d
Of all the remarkable intellectual achievements of
ancient Greek civilization, none has had greater
subsequent impact than the developnient or the axiomatic ~netliodof analysis. N o serious traces are to be
found in the earlier civilizations of Babylon, China,
Egypt, or India. The exact history of the beginnings is
not known. but elements that can now be identified
emerged in the fifth century BC. A good reference is
Knorr ( 1975). What can be said, and is important for
Other early examples of axioniatic theories aimed at
empirical matters can be found in the large medieval
literature on qualitative axioms for the nieasurement
of weight (Moody and Clagett 1952). Even more
impressive examples are to be found in the medieval
literature on physics. Some of the most s ~ ~ b tand
le
interesting work is that of'Nicole Oresnie ( 1 968) in the
fourteenth century. (See especially his treatise,
Tractatus I
Cor~figrrr~tioriiA~.rsQlrulittrtutn
M O I L I L I I ~What
~ . ) is surprising is Oresme's ~ l n e x ~ e c t e d l ~
subtle approach in a geometrical Famework to the
phenomena of intensive qualities and ~ n o t i o ~ iAn
s.
Axiomatic Theories
example for Oresme would be that of a surface being
more o r less white.
Finally, in the geometric axiomatic tradition of
analyzing phenomena in natural science, the two great
late examples of deeply original work were the
seventeenth-century treatises of Huygens' The Pendulum Clock (Huygens 1673/1986). And, of course, as
the' second example, Newton's Principia (Newton
168711946). Both Huygens and Newton formulated
their axioms in the qualitative geometric style of Euclid
and other Greek geometers two thousand years earlier.
1.1 Axiomatic Geometr'y in the Nineteenth Century
Without question, development in axiomatic methods
in the nineteenth century was the perfection and
formalization of the informal Greek methods that had
dominated for so many centuries. The initial driving
force behind this effort was certainly the discovery and
development of non-Euclidean geometries at the
beginning of the century by Bolyai, Lobachevski, and
Gauss. An important development later in the century
was the discovery of Pasch's (1882) axiom a s a
necessary addition to the Euclidean formulation.
Pasch found a. gap in Euclid which required a new
axiom, namely, the assertion that if a line intersects
one side of a triangle, it must also intersect a second
side. This was just the beginning for Pasch. H e created
the modern conception of formal axiomatic methods,
which has been the central aspect of the model for
axiomatic work up until present times. The axiomatic
example in geometry that had the most widespread
influence was Hilbert's Grundlagen der Geometrie, first
published in 1897 and still being circulated in later
editions (Hilbert 1956).
2. Ingredients oJSrandard Axiomaric Practice
Building, especially, o n the work in the later part of the
nineteenth century in axiomatizing geometry, early in
the twentieth century there was widespread axiomatization in mathematics and, to a lesser extent, in the
empirical sciences. The main ingredients of the methods for axiomatizing theories were the following:
statement of the primitive concepts of the theory,
statement of the prior mathematics basis assumed,
statement of the axioms, characterization of models of
the theory and a definition for isomorphism of two
such models, proof of a representation theorem when
Possible, and, finally, some analysis of invariance of
the models of the theory. Before turning to an explicit
discussion of these ingredients, it is worth noting that
the emphasis should really be o n models of the theory,
which is what axiomatizing a theory makes clear. F o r
ltis models of the theory, i.e., structures which satisfy
the theory, as explained in Sect. 2.3;thal exhibit the
nature of the theory, whether it be in geometry,
economics, psychology or some other science. (For an
elementary, but detailed, account of the concepts
discussed in this section, see Suppes (195711999,
Chaps. 8 and 12).)
2.1
Primitive Concepts of a Theory
The first point t o recognize in axiomatizing a theory is
that some concepts are assumed as primitive. Their
properties are to be stated in the axioms, and,
therefore, it is important to know how many such
concepts there are and what is their general formal
character, as relations, functions, etc. In Hilbert's
axioms for elementary plane geometry, the five primitive concepts are those of point, line, plane, betweenness, and congruence. In contrast, in psychological
theories of measurement, a n ordering of the stimuli or
other phenomena is almost always assumed-in this
case, a weak ordering; that is, a binary relation that is
transitive and connected, rather than the geometric
ordering of betweenness. In addition to a primitwe
concept of ordering, there are other relations, for
example, a primitive concept of the comparisoil of
stimulus differences o r an operation of combination
for extensive measurement, as in the case of subjective
probability. These measurement examples also apply
to a large literature on utility and subjective probability in economics. In psychological theories of
learning, even of the simplest nature, we would need
such concepts a s that of stimulus, response and a
conditioning relation between stimulus and response.
Theories without much more elaboration would
require the important concept of similarity or resemblance, in order to develop a theory of the
fundamental psychological phenomena of generalization, discrimination a n d transfer.
There are also important theories that use much
simpler primitive concepts. A good example would be
the theory of zero-sum, two-person games in normal
form. The primitive concepts are just that of two
nonempty sets X and Y that are arbitrary spaces of
strategies for the two persons and a numerical function
M, defined on the product space X x Y. The intuitive
interpretation of M is that it is the payoff or the utility
function for the player whose space is X. The negative
of that is the payoff for the player whose space is Y.
Later a representation theorem is given for finite
games of this sort.
2.2 Axioms as Defining Theories
As is widely recognized, axioms are what intuitively
characterize a theory, whether it be of geometry, game
Axiomatic Theories
theory o r a psychological theory of measurement.
From a formal standpoint, something more can be
said. The essence of the matter is that to axiomatize a
theory is to define a certain set-theoretical predicate.
This is'just as valid in the empirical sciences as in pure
mathematics. The axioms, of course, are the most
important part of such a definition.
Because the theory of weak orderings is used widely
in both economics and psychology, as part of the
theory of choice, it will be useful to give a formal
definition. and, therefore, the axioms for a weak
ordering.
Definition 1 Let A be a nonempty set and R abinary
relation on A, i.e.,let R be a subset of the product space
A x A. A structure ( A , R ) is a weak ordering ifand only
if the following two axioms are satisjied, for every a , b
and c in A:
Axiom 1 R is transitive on A, i.e., i f a R y and bRc
then aRc.
Axiom 2 R is connected on A, i.e., aRb or bRa.
There are various general methodological and folklore recommendations about the way in which axioms
should be written--clarity, lack of ambiguity and
simplicity are familiar. More substantive results about
the form of axioms are discussed in Sect. 4 on
first-order formalization. There are many definite
mathematical results about the form of axioms, some
of which are also discussed later.
Independence of Axioms
An early and important substantive recommendation
is that the axioms be independent; i.e., none can be
derived from the others. T h e question then arises from
a methodological standpoint: How is independence to
be established? The answer is in terms of models o f the
axioms. T o make the matter still more explicit, we
consider possible realizations of the theory. These are
set-theoretical structures that have the right form. For
example, in the case o f weak orders, a possible
realization is an ordered pair consisting o f a nonempty
set and a blnary relation o n that set. Such an arbitrary
pair is called a possible realization because a s a
possible realization it is not a m o h l of the theory
unless the axioms of the theory are also satisfied. An
obvious example of a model of the theory of weak
ordering is the pair consisting of the set of positive
integers and the relation of weak inequality 2 .
The independence of a given axiom of a theory is
established by finding a possible realization of the
theory in which all the axioms, except the particular
one in question, are satisfied. The deductive argument
to show that this yields a proof of independence is
intu~tivelyobvious and will not be made In a more
formal manner here. But the essence is that if the
concept were not independent, but definable, then it
would necessarily hold in the' model, just like the
remaining axioms, and so a contradiction of its both
2.3
holding and not holding in the given model would be
obtained. So, by reductio ad absurdum it must be
independent. It will be useful to consider an example
or two.
To show that the axiom of connectedness for weak
orders is independent of the axiom of transitivity, it is
necessary to take a possible realization that is also a
transitive relation. So, let A = { l , 2, 3) and R =
{l, 2). In other words, R is just the strict numerical <
relation on the subset {I, 2) of A . Then it is obvious
that this relation is transitive but not connected for the
number 3, which is in the set A, and does not stand in
the relation R to any other element in A . T o show that
transitivity is independent of connectedness, that is,
Axiom 1 for weak orders is independent of Axiom 2,
it is sufficient to take the same set A as before, but now,
the relation R is the set of the ordered pairs {(I, 2),
(2,3), (3, I), (1, I), (2, 2), (3, 3)). Then it is clear that
any two elements in the set A are connected by the
relation R, but the relation R is not transitive, for, in
order to be transitive, it must also have the pairs (I, 3):
(2, 1) and (3, 2).
The examples of independence given are trivial, but
it is to be emphasized that it can often be a difficult
task to establish whether o r not an axiom is independent of the remaining axioms; that is, whether or
not it can be derived from the remaining axioms. A
very familiar mathematical example, with a long
history in the twentieth century, is the proof that the
axiom of choice is independent of the other standard
axioms of set theory.
2.4
Padoa's Principle for Proving Independence q/
Primitive Concepts
Less familiar than proving the independence of axioms
is the method of using models of a theory to prove
independence of the piimitive concepts of the theory.
To prove that a particular primitive concept of a
theory, for. -example, the notion of congruence in
Euclidean geometry, is independent of the other
primitive concepts, it is sufficient to find two models of
the theory, such that the domain of both models is the
same, the two models are the same for all the other
primitive concepts of the theory, but the two models
differ only in their realization of the concept in
question. Thus, to prove congruence independence,
what is required are two different models of the axiom
in question, which are the same for all the other
concepts, such as point, line, etc., but which have two
distinct notions of congruence.
In the case of weak orders, Padoa's principle can be
used in an obvious way to show that the concept of the
binary relation is independent of the given set A. It
suffices, for example, to use the two different orderings
<, 2 on the set of positive integers. On the other
hand, since R is connected,-the set A is definable in.
terms of the relation R for the special case of weak
orderings by taking the union of the domain and range
of tlie relation. But for diKerelAltorderings, for example, partial orderings, which are reflexive, antisymmetric and transitive on the set A, it is easy to show
that the set A is an independent concept. Just let A in
one model be a proper subset in tlie other, and elemen~s
in tlie relation come only from the first set.
The separation of the purely set-theoretical characterization of the possible realizations of a theory from the
axioms proper, which characterize models of a theory,
is significant in defining certain important concepts.
For example, the notion of two models of a theory
being isolnorphic is often said to be axiom-free, since
the definition of isomorphism of the models of a
theory really depends on just the set-theoretical characterization of the possible realizations. A satisfactory
general definition of isomorphism for the structures
that are possible realizations of any sort of theory is
difficult, if not impossible, t o formulate. The usual
practice is to formulate a special definition for the
possible realizations of some particular theory. This is
what will be done here, as already illustrated in the
case of binaryrelations. A possible realization is a settheoretical structure that is a nonempty set and a
binary relatioii whose domain and range are included
in that set. In this case the definition of isomorphism is
as follows.
Definition 2 A binary relalion sfructure ( A , R ) is
isornorpl~icto a binary relalion structure ( A ' . R ' ) if,
and O ~ ~ lf,J J[here is a function f such that
( a ) D(./')
= A and R ( f ) = A'
( b ) f is a one-one.fi~n.ction
( c ) For a , b in A , aRb ifand only i f f ( a ) R ' f ( b ) .
The definition of isoinorphis~nfor possible realizations of a theory is used to formulate a representation
theorem, which has the following meaning. A certain
class of structures or models of the axiomatized theory
is distinguished for some intuitive or systematic reason
and is shown to exeinplify within isomorphism every
other model of the theory. This means that, given any
model of the theory, there exists in this distinguished
class an isomorphic model. A good example of this can
be found in the axiomatic theory of extensivemeasurement. Given any empirical structure that represents
the data about subjective judgments of probability
and satisfies the axioms of the theory, there is a
flunierical structure satisfying the axionis that is
lsoniorphic to the empirical structure. Note, of course,
that there is not any one single such isomorphism.
Different individuals can have different empirical
structures realizing their subjective probabilities, but
there will be for each of the111 a particular numerical
model of the axioms that is isomorphic to each given
empirical structure.
Here is a simple and obvious, but. usef~~l.,
exampleof
a representation result for two-person, zero-sulii
gamnes that are finite. A-game(X, Y, M ) , as introduced
earlier, is finite just in case tlie sets X and Yare finite.
So., with the definition of isoinorphism obvio~is
froin the result now to be stated, ally finite game
G (X, Y, M ) w i t h X = (I,,
..., x,,,)and Y = (I:,, ....y,)
is isoinorphic to the game G' = (I,,? I,,. M ' ) , and
M ( i ,j] = Mixi,yj), where the notation I, denotes the
set of positive integers 1, ..., k . Therefore, any finite
two-person zero-sum game may be represented by a
game where X and Y are initial segments of' the
positive integers, even if vivid substantive descriptions
of the individual strategie.~sf and v j have been given.
-
3.6
h7variance Theoren~s
for Asion.~ati~ed
Theories
In addition to having representation theorems for the
models of a given axiomatized theory, it is also
significant and useful to have explicit invariance
theorems. The intuitive idea of invariance is most
naturally explained either in terms of geometric
theories o r measurement theories.
Given a class of models, for example, analytical
models in geometry and numerical models in measurement, w ~ t hrespect to which any other model of the
theory can be represented, the invariance theorem is
a n expression of how unique is the representation in
terms ol' such an analytic model. For example, in the
case of the representation theorem for axiomatic
formulations of Euclidean geometry, the invariance
theorem states that any analytic model is unique only
up to the group of Euclidean motions. Thismeans that
for any analytic representation of Euclidean geometry,
any reflections, rotations o r translations of the analytic
model will produce a new analytic model, also isomorphic to the set-theoretical model satisfying the
qualitative synthetic axioms of Euclidean geometry. In
fact, the most general theorem widens the scope to
what is often called the group of generalized Euclidean
transformations, namely, those that also permit a
change in scale, so that no unit of measurement is
fixed.
The situation is very similar in the case of theories of
measurement. Given an empirical structure and a
representing nunierical model isomorphic to that
structure, then, in the case of intensive quantities such
as cardinal utility. for example. the numerical model is
unique only up to the group of affine transformations;
that is, transformations that multiply the unit o r
measurement by a positive real number and add a
constant as well. This means that in f~~iidamental
measurement of a cardinal utility there is nothing but
convention in the choice of a unit of measurement or
a zero value.
Classical examples of important invariant theorems in physics are that structures of classical physics
Axionzntic Theories
are invariant up to Galilean transformations, and in
special relativity, invariant up to Lorenz transformations-in
both these cases, further generalizations
arise .by also permitting changes in the units of
measu remen t .
4.
Theories with Stnndard Fortnaliration
The set-theoretical framework for axiomatizing tlieories just discussed is used implicitly throughout the
mathematical social sciences. Most of the axiomatic
work that takes place in the social sciences can be put
in a straightforward way within the framework just
described. On the other hand, there is a tighter
framework for discussing the axiomatization of theories that leads to clarification of problems that arise in
the conceptual or qualitative thinking about theories
in the social sciences. Tlie purpose of this section is to
describe in an informal way this more narrowly defined
framework and to give a sense of the kind o f results
that can be obtained. Most of the results are of a
negative sort, important because they show what
cannot be done by apparently very natural qualitative
formulations of theories of preference, of subjective
probability or related kinds of qualitative measurement, or scaling problems, especially in econo~nicsand
psychology. but also in anthropology, political science, and sociology.
A language with standard formalization is a
language that is given a precise formulation within
first-order logic. Such a logical framework can be
characterized easily in an informal way. This is the
logic that assumes:
(a) one kind of variable;
(b) logical constants, mainly the sentential connectives such as and ;
(c) a notation for the universal and existential
quantifiers; and
(d) the identity symbol =.
A language formulated within such a logical framework is often called an elementary language. Ordinarily, three kinds of nonlogical constants occur in
axiomatizing a theory in such a language-the relation
symbols, also called predicates, the operation symbols
and the individual constants.
The grammatical expressions of the language are
divided into terms and formulas, and recursive definitions of each are given. The simplest terms are
variables or individual constants. New terms are built
up by combining simpler terms with operation syinbols in a recursive manner. Atomic formulas consist o r
a single predicate and the appropriate number of
terms. Compound formulas are built up from atomic
f'orinulas by means of sentential connectives and
quantifiers.
Possible realizations of elementary theories, i.e.,
theories formulated in an elementary language, assume an especially simple form. First, there is a
nonempty domain for the structure. Second, cor-
responding to any relation syn~bolof the theory, there
is a corresponding relation, including sets representing
predicates as one-place relations. Corresponding to
any operation symbols in the theory are operations
defined on the domain of the structure, and, finally,
individual objects of the domain correspond to the
individual nonlogical constants of the theory. I t is
worth noting in passing that the definition of isomorphism for such elementary structures is straightforward and simple, which is not always the case for
theories formulated in more con~plicatedset-theoretical languages.
4.1 Some Positive Resrllts About Axiomatizrrbilit~v
The first positive result uses two concepts. First, the set
of all the finite models of a theory is called a,finirary
class of elementary structures. Second. a theory is
recursively axiomatizable when there is an algorithm
for deciding whether or not any formula of the
language is an axiom of the theory in question.
Theorem 1 Any finilnry class of t ~ ~ o d e lofs an
elemenlury theory is axiornatizable, bur no[ necessarih
recursive1.v axiornatizable.
The importance of this result is showing that the
expressive power of elementary languages is adequate
for finitary classes but not necessarily f'or the stating or
a set of recursive axioms.
A more special positive result about finitary classes
of models can also be given:
. ~ ofn~easureTheorem 2 Lel K be ~ h e f i i i t u r class
Itlent structures wilh respect to an elen~en~ary
lar~gl~age
L and wit11respect lo a nun~ericaltnodel9 of L suc11thut
K iticludes all finite models of L hon-~or~~orpliicall~
embeddable ill W.If the domain, relritiotis, fi~nctions,
and corlstants o f ' g are dejnable in elen~enraryform in
rertns of ( R e , <.+ , . ,0, I ) [hen the set ofsentences of
L [hat are suti.yfied in every model of K is recursively
a.xion~atizable.
A urliuersal senlencr is one that has only universal
quantifiers at the beginning with the scope of a
quantifier the remainder of the sentence. In practice,
such sentences are written as quantifier-free statements
to simplify the notation. The axioms of weak ordering
given earlier are of this kind. T o be explicit, the
conjunction of the two given axioms is a universal
sentence? and this single universal sentence is, in this
form, the single axiom for a weak ordering.
Theorem 3 ( Vaughl 's Crilerion 1954). Let L be an
elementarj~latlgtlnge ~ ~ i t h ofitnction
ut
s ~ ~ r ~ b oAl finis.
tc11.y class K of n~easurementstrucleires ( with respect to
L ) is asion-latizable 17)! (1 t~niuersalsetitet~ce iff K is
closed t~ndersul>models and there is at1 it~teger11 such
[hat if any ,finite rnodel M of L has the pt.operty that
etler.y subtnodel of M with tlo 1110re t11a11n e l e t i ~ e ~i ~. in
~t . ~
K, then M is in K.
The intuitive idea ol'vaught's criterion for finitary
classes of models of a theory is easy to explain.
Consider again weak orderings. Because the axioms
A.riorncitic Theories
involve just three distinct variables. it is sufficient to
chcck triples oT objects in the domain of an e~npirical
structure to determine if it is a model of the theory.
Generally speaking, the number oT distinct variables
determines the size of the submodels that must be
checked to see if universal axioms are satisfied. T o
have a universal axiom Tor a theory, or what is
equivalent. a finite set of universal axioms, it is
necessary that the number oT distinct variables be
some definite number, say n. By examining subnlodels
involving no more than r l objects, i t is then sufficient to
determine satisfaction of the axiom or axioms, and
this is the intuitive idea of Vaught's criterion.
4.2
-
. Tl~erzthe finitury c1a.s~J of'/neasure~nen/strlrc/lrre.s
,for the inclis/ingui.shibili/y relcition
is not tixionzatizable in L b y ci uniz~erscrlserzterlce.
The next case oT a negative result applying the
negation oT Vaught's criterion (Theorem 3) is the
proof that the qualitative theory of utility differences
or the qualitative theory, more generally, of various
psycholnetric sensations is not axiolnatizable by a
universal sentence, contrary, of course, to the simple
thcory of order. This result is due to Scott and Suppes
(1958).
Consider the elementary language whose only symbol is the quaternary relation symbol 'D' with thc
intended numerical interpretation
-
Sor?le N e g c ~ l i ~Results
e
About Axiomatizubility
To begin with, it isuseful to state as a theorem what is
a corollary of Vaught's criterion. It is simply the
negation of it for determining when a finitary class of
models oT a theory is not axiomatizable in an elementary language by a universal sentence.
Theorem 4 Let L be an elementarv lunguage
without function .synzbol.r and let K be afinitary class of
measuren1ent structures (with respect to L ) closedunder
suhmor1el.s. Tl?eri K is not axion~atizableby a urzh~ersal
sen rence u / L ; n o r every integer,11 there is ajinite model
M r f l L that kns the property /hat every sl~bmodelwith
no niore thaiz n elenzents is in K , but M is M O I in K.
It is natural to interpret the negative result stated in
the general form here as showing that the complexity
oT relationships in finitary classes satisfying the hypotheses of the theorem is unbounded, at least
u~lboundedwhen the theory must be expressed by
quantifier-free formulas of elementary logic.
The first application is something rather surprising.
A semi-order is a structure (A, >) satisfying the
following three axioms.
Axiom 1 It is not the case that ( n > a).
Axiom 2 If ( a > b & a' > b') then ( a >
b' v a' > b).
Axiom3 I f ( a > b & b > c ) t h e n ( a > d v d > c ) .
When the set A is finite, the following numerical
representation holds:
for every a and b in A, a > b
if and only if f ( a ) > f ( b ) + I.
What is now surprising is a result about the
indistinguishability relation for semiorders', that is, the
that is the negation of the characterization
relation
of the semiorder, namely, for a and b in A
-
The following theorem is due to Roberts (1969).
Theorem 5 Let L be the elemei~tarylanguage whose
on!v nonlogicnl symbol is the binary relationnl symbol
We then define the finitary class of measurement
structures for algebraic difference as consisting of all
models (A, D) such that
(a) A is a nonempty finite set;
(b) D is a quaternary relation on A; and
(c) (A, D) is ison~orphicto (A', A), where A' is a
finite set of numbers and A is the quaternary numerical
relation such that for real numbers x, y, u and v,
x.vAuv iff .K -y 2 t~- c.
Theorem 6 Let L be the elementary language whose
orzlj~nor~logicalsyn~bolis tlze quaternaty relation symbol
D. The finitary class 9 of mensurernent structuresjor
(algebraic) dlffcrence is not axiomatiznble in L
a
universal sentence.
The intuitive idea of the proof can be seen from
construction of a ten-element structure, all of whose
substructures have a numerical representation, but
which does not itself have such a representation. The
idea of this construction can then be generalized for
arbitrary n in order to apply Theorem 4.
Using the same ideas, Titiev (1972) extended the
results of Theorem 6 to additive conjoint measurement and to multidimensional scaling with the
Euclidean metric. Titiev (1980) also gives a negative
proof for the city block metric for n < 3 dimensions.
Using more sophisticated logical results, it is possible
to extend most of the results just stated to not being
finitely axiomatizable. This means that ex~stential
quantifiers can be introduced, but the number of
axioms must be finite in character. The main results
here are due to Per Lindstrom, which together with the
other results mentioned in this section are presented in
detail in Luce et al. (1 990).
Theorem 7 The finitary clnss of rnensurer?leirt
srructllres for algebraic d~ferenceis not ,finitely axiomatizable in ~lze elernentnrv language whose only
nonlogical sjlmbo/ is the quarerilnry relation syinbol D.
In a way, a still stronger negative result about
axiomatiiability can be proved for Archimedean
axioms. Here is one standard formulation of such an
b the
axiom. Ifn 3 b then for some n, nb a, where i ~ is
combination o f n copies of b. Notice that it is necessary
>
Axion~aticTheories
to introduce a quantifier for integers not just for the
empirical objects of the domain. Because the exact
formulation of the negative result is rather COIIIplicated, the following informal theorem is given.
Theorem 8 For any standard elenzentory language
used to,forniz4late a theory of measurement there is no
set o f elernentc1r;v forrnulas of the language equivalerzt to
an Archimedean asion1 for the theory.
The best way to think about the Archimedean axiom
in this context is that it is a second-order axiom and,
therefore, cannot be formulated by an equivalent set
of formulas, in first-order logic.
Still another way of looking a t this result is that to
characterize the real numbers we need some sort of
second-order axiom such as Dedekind completeness,
the Cauchy sequences o r the least-upper bound axiom,
but none of these axioms, including the Archimedean
axiom, can be formulated in a first-order language
whose variables take real numbers as values. Narens
(1974) provides a general account of Archimedean
axioms in various forms.
For deeper and more general results on axiomatizability there is much recent work that can be
cited, especially concerning the definability of linear
orders for classes of finite models and the problem of
the complexity of the class (Stolboushkin 1992,
Gurevich and Shelah 1996, Hetla et al. 1997).
See also: Mathematical Models in Philosophy of
Science; Measurement, Representational Theory
of; Measurement Theory: Conjoint; Measurement
Theory: History and Philosophy; Ordered Relational
Structures
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